lucid.rodeo · incerto · barrier

The absorbing barrier

Ruin is not a bad outcome; it is the end of outcomes. Above: the arithmetic of repeated exposure, exact. Below: six hundred walkers and a floor that keeps what it touches — the flattering average of the survivors, measured live against the whole population’s zero.

Exact Panel one is the closed-form curve (1−p)ⁿ — no simulation at all. Panel two is real seeded ±1 walks with absorption checked at every step; its one smooth overlay is the diffusion approximation erf(B/√(2n)), labelled as exactly that — at B = 20, n = 10,000 it differs from the exact discrete answer only in the fifth decimal (0.15852 vs 0.15851). the fence →

The repeat-exposure curve

survival at 365 (1 year daily)
at 1,000
at 10,957 (30 years daily)
ruin within the 30 years
axes x 1→10⁵ (log; ticks 1, 10, 10², 10³, 10⁴, 10⁵) · y 0→1 (ticks 0, 0.5, 1) · marks at 365, 1,000, 10,957
 

The walk

step
alive (of 600)
absorbed
survival (sim)
continuum approx erf(B/√(2n))
mean endpoint, survivors
mean, all 600 (dead frozen at −B)
seed
axes walk y −20→+ (0 marked) · x 0→10,000 (ticks 0, 5,000, 10,000) · strip y 0→1 (ticks 0, 1)
 

The upstairs arithmetic is Taleb’s ruin argument, paraphrased: a per-round ruin probability that looks negligible compounds across rounds as (1−p)ⁿ, and because ruin is absorbing — the player is removed from every later round — repetition converts small probabilities into near-certainty. In Skin in the Game (Book 8, ch. 19, “The Logic of Risk Taking”) he separates ensemble probability — a hundred players on one day — from time probability — one player on a hundred days — and argues that once ruin is on the table, ordinary cost–benefit analysis stops applying. The Precautionary Principle paper makes the same move at systemic scale: when a potential harm is fat-tailed and the barrier irreversible, the question is not the odds of one round but the certainty of repetition. And the two-averages page quietly assumes this page away: the time-average growth rate g(f) is the limit of (1/n)·log(Wₙ/W₀), which presupposes wealth stayed positive at every step — log 0 is not a low growth rate but the end of the quantity being measured. The smooth Kelly model avoids ruin by construction (infinitely divisible bets, bounded losses); applying its conclusions where bets are discrete imports an unearned no-ruin assumption.

Downstairs, survivorship bias is measured rather than asserted. A stopped symmetric walk is still a martingale, so the full population — the dead counted where they froze, at the floor — has mean exactly zero at every step. Yet the walks that happen to miss the barrier average about +106 at n = 10,000 with B = 20 (exact value 106.17 by reflection-principle binomial sums — roughly one full standard deviation, since √n = 100). Nobody cheated; the graveyard was simply left out of the average — the same books the turkey kept, survivors-only, until it wasn’t one. The dashed overlay is the diffusion (Brownian) approximation erf(B/√(2n)): exact only in the continuum limit, though at these parameters the exact discrete survival 0.15851 and the approximation 0.15852 agree to four significant figures. One drawing confession: paths are plotted from every-25th-step samples for legibility, while absorption is checked at every one of the 10,000 steps.

Sources, fetched and verified: Nassim Nicholas Taleb, Skin in the Game: Hidden Asymmetries in Daily Life (Random House, 2018), Book 8, ch. 19, “The Logic of Risk Taking” — essay version; Taleb, Read, Douady, Norman & Bar-Yam, “The Precautionary Principle (with Application to the Genetic Modification of Organisms)”, arXiv:1410.5787 (2014); Ole Peters, “The ergodicity problem in economics”, Nature Physics 15 (2019): 1216–1221, doi:10.1038/s41567-019-0732-0 (paywalled; linked for the record). All prose here is paraphrase — none of the living authors’ sentences are reproduced.

the wing →